Tag - Infinite groups

Waldemar Hołubowski: Normal subgroups in the group of column-finite infinite matrices

7th June, 2021

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GLn(K) (K a field, n ≥ 3) which is not contained in the centre, contains SLn(K). A. Rosenberg gave description of normal subgroups of GL(V), where V is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the centre and the group of linear transformations g such that g − idV has finite-dimensional range the proof is not complete. We fill this gap for countable-dimensional V giving a description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field. Similar results for Lie algebras of matrices will be surveyed.

Lancelot Semal: Unitary representations of totally disconnected locally compact groups satisfying Ol’shanskii’s factorization

5th June, 2021

We provide a new axiomatic framework, inspired by the work of Ol'shanskii, to describe explicitly certain irreducible unitary representations of second-countable non-discrete unimodular totally disconnected locally compact groups. We show that this setup applies to various families of automorphism groups of locally finite semiregular trees and right-angled buildings.

Robert Kropholler: Groups of type FP2 over fields but not over the integers

10th May, 2021

Being of type FP2 is an algebraic shadow of being finitely presented. A long standing question was whether these two classes are equivalent. This was shown to be false in the work of Bestvina and Brady. More recently, there are many new examples of groups of type FP2 coming with various interesting properties. I will begin with an introduction to the finiteness property FP2. I will end by giving a construction to find groups that are of type FP2(F) for all fields F but not FP2(ℤ).

Yago Antolin: Geometry and Complexity of positive cones in groups

10th May, 2021

A positive cone on a group G is a subsemigroup P such that G is the disjoint union of P, P−1 and the trivial element. Positive cones codify naturally G-left-invariant total orders on G. When G is a finitely generated group, we will discuss whether or not a positive cone can be described by a regular language over the generators and how the ambient geometry of G influences the geometry of a positive cone.

Benjamin Fine: Elementary and universal theories of group rings

6th May, 2021

In a series of papers the above authors examined the relationship between the universal and elementary theory of a group ring R[G] and the corresponding universal and elementary theory of the associated group G and ring R. Here we assume that R is a commutative ring with identity 1 ≠ 0. Of course, these are relative to an appropriate logical language L0, L1, L2 for groups, rings and group rings respectively. Axiom systems for these were provided. Kharlampovich and Myasnikov, as part of the proof of the Tarskii theorems, prove that the elementary theory of free groups is decidable. For a group ring they have proved that the first-order theory (in the language of ring theory) is not decidable and have studied equations over group rings, especially for torsion-free hyperbolic groups. We examined and survey extensions of Tarksi-like results to the collection of group rings and examine relationships between the universal and elementary theories of the corresponding groups and rings and the corresponding universal theory of the formed group ring. To accomplish this we introduce different first-order languages with equality whose model classes are respectively groups, rings and group rings. We prove that if R[G] is elementarily equivalent to S[H] then simultaneously the group G is elementarily equivalent to the group H and the ring R is elementarily equivalent to the ring S with respect to the appropriate languages. Further if G is universally equivalent to a nonabelian free group F and R is universally equivalent to the integers ℤ then R[G] is universally equivalent to ℤ[F] again with respect to an appropriate language. It was proved that if R[G] is elementarily equivalent to S[H] with respect to L2, then simultaneously the group G is elementarily equivalent to the group H with respect to L0, and the ring R is elementarily equivalent to the ring S with respect to L1.

The structure of group rings is related to the Kaplansky zero-divisor conjecture. A Kaplansky group is a torsion-free gorup which satisfies the Kaplansky conjecture. We next show that each of the classes of left-orderable groups and orderable groups is a quasivariety with undecidable theory. In the case of orderable groups, we find an explicit set of universal axioms. We have that 𝒦 the class of Kaplansky groups is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in 𝒦 or more generally two torsion-free groups are universally equivalent.

Finally we consider F to be a rank 2 free group and ℤ be the ring of integers. we call ℤ[F] a free group ring. Examining the universal theory of the free group ring ℤ[F] the hazy conjecture was made that the universal sentences true in ℤ[F] are precisely the universal sentences true in F modified appropriately for group ring theory and the converse that the universal sentences true in F are the universal sentences true in ℤ[F] modified appropriately for group theory. We prove that this conjecture is true in terms of axiom systems for ℤ[F].

Mikhail Belolipetsky: Growth of lattices in semisimple Lie groups

22nd April, 2021

A discrete subgroup G of a Lie group H is called a lattice if the quotient space G/H has finite volume. By a classical theorem of Bieberbach we know that the group of isometries of an n-dimensional Euclidean space has only finitely many different types of lattices. The situation is different for the semisimple Lie groups H. Here the total number of lattices is infinite and we can study its growth rate with respect to the covolume. This topic has been a subject of our joint work with A. Lubotzky for a number of years. In the talk I will discuss our work and some other more recent related results.

Laura Ciobanu: Free group homomorphisms and the Post Correspondence Problem

19th April, 2021

The Post Correspondence Problem (PCP) is a classical problem in computer science that can be stated as: is it decidable whether, given two morphisms g and h between two free semigroups A and B, there is any non-trivial x in A such that g(x)=h(x)? This question can be phrased in terms of equalizers, asked in the context of free groups, and expanded: if the 'equalizer' of g and h is defined to be the subgroup consisting of all x where g(x)=h(x), it is natural to wonder not only whether the equalizer is trivial, but what its rank or basis might be. While the PCP for semigroups is famously insoluble and acts as a source of undecidability in many areas of computer science, the PCP for free groups is open, as are the related questions about rank, basis, or further generalizations. However, in this talk we will show that there are links and surprising equivalences between these problems in free groups, and classes of maps for which we can give complete answers.

Tullio Ceccherini-Silberstein: Linear cellular automata, linear subshifts, and group rings

25th March, 2021

Let G be a group and let V be a finite-dimensional vector space over a field K. We equip VG = {x : G → V} with the prodiscrete uniform structure, the G-shift action ((gx)(h) := x(g-1h)), and the natural structure of a K-vector space. A G-invariant closed subspace X ≤ VG is called a linear subshift. A linear subshift X ⊆ VG is said to be of finite type provided that there exists a finite subset Ω ⊆ G and a subspace W ⊆ VΩ such that

X = X(Ω,W) := {x ∈ VG : (gx)|Ω ∈ W for all g ∈ G}.

The group G is said to be of linear-Markov type if for every finite-dimensional vector space V over any field K, every linear subshift X ⊆ VG is of finite type. A uniformly continuous and G-equivariant K-linear map τ : VG → VG is called a cellular automaton. The group G is said to be linearly surjunctive provided that for every finite-dimensional vector space V over any field K the following holds: every injective linear cellular automaton τ : VG → VG is surjective.

THEOREM 1 (CS-Coornaert 2007) Sofic groups are linearly surjunctive.

COROLLARY 1 (Elek-Szabo 2004; CS-Coornaert 2007) Group rings of sofic groups are stably finite.

THEOREM 2 (CS-Coornaert-Phung 2020) A group is of linear-Markov type if and only if the group ring K[G] is left-Noetherian for any field K.

COROLLARY 2 (CS-Coornaert-Phung 2020) Polycyclic-by-finite groups are of linearly-Markov type.

Arman Darbinyan: Subgroups of left-orderable groups

18th March, 2021

A recent advancement in the theory of left-orderable groups is the discovery of finitely generated left-orderable simple groups by Hyde and Lodha. We will discuss a construction that extends this result by showing that every countable left-orderable group is a subgroup of such a group. In conjunction with this construction, we will also discuss computability properties of left-orders in groups. Based on a joint work with M. Steenbock.

Giles Gardam: Kaplansky’s conjectures

11th March, 2021

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and finish with my recent counterexample to the unit conjecture.